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What is the number of ways of partitioning a positive number $k\leq mn$ using non-increasing parts such that the number of parts can be at most $n$ and the value of each part can be at most $m$?

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You are asking for the coefficients of the Gaussian or $q$-binomial coefficient. If $$ [n] := \frac{q^n-1}{q-1} $$ and $$ [n+1]! := [n+1]\cdot [n]!, \quad [0]!=1 $$ then you want the coefficient of $q^k$ in the $q$-binomial coefficient $$ \frac{[m+n]!}{[m]![n]!}. $$ There is no explicit formula. Googling on Gaussian binomial should lead to a proof.

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